In this paper, we calculate terms of associated polynomials of Perrin and Cordonnier numbers by using determinants and permanents of various Hessenberg matrices. Since these polynomials are general forms of Perrin and Cordonnier numbers, our results are valid for the Perrin and Cordonnier numbers.

Arithmetical sequences for the exponents of composite Mersenne numbers are obtained from partitions into consecutive integers, and congruence relations for products of two Mersenne numbers suggest the existence of infinitely many composite integers of the form 2p − 1 with p prime. A lower probability for the occurrence of composite Mersenne numbers in arithmetical sequences is given.

A new type of Fibonacci sequence is introduced and explicit formulas for the form of its members are formulated and proved. It is an extension of the special Fibonacci sequence, introduced in and called a Pulsated Fibonacci sequence.

In this paper, we establish two modular relations for the Rogers–Ramanujan–Slater functions of order fifteen. These relations are analogues to Ramanujan’s famous forty identities for the Rogers–Ramanujan functions.
Furthermore, we give interesting partition theoretic interpretations of these relations.

It is well known that, the problem of finding a sequence of real numbers an, n = 0, 1, 2, …, which is both geometric (an + 1 = kan; n = 0, 1, 2, …) and balancing (an + 1 = 6an − an − 1, a0 = 0, a1 = 1) admits an unique solution. In fact, the sequence is 1, λ1, λ12, …, λ1n, … where λ1 = 3 + √8 satisfies the balancing equation λ2 − 6λ + 1. In this paper, we pose an equivalent problem for a sequence of real, nonsingular matrices of order two and show that, this problem admits an infinity of solutions, that is there exist infinitely many such sequences.

A generalization of Ripà’s square spiral solution for the n × n × … × n Points Upper Bound Problem. Additionally, we provide a non-trivial lower bound for the k-dimensional n1 × n2 × … × nkPoints Problem. In this way, we can build a range in which, with certainty, all the best possible solutions to the problem we are considering will fall. Finally, we provide a few characteristic numerical examples in order to appreciate the fineness of the result arising from the particular approach we have chosen.

This paper presents an original composition based on Fibonacci numbers, to explore the inherent aesthetic appeal of the Fibonacci sequence. It also notes the use of the golden ratio in certain musical works by Debussy and in the proportions of violins created by Stradivarius.